Induction logging has provided important measurements of formation conductivity for many years. Conventional induction logging tools use coils that have magnetic moments with their axes aligned with the tool axes. Techniques have been developed and employed for relatively accurate compensation for borehole effects in these tools. More recently, some induction logging tools employ antennas with magnetic dipole moments oriented in both the longitudinal (i.e., axial) direction, and transverse directions.
Modern induction logging tools, such as the 3-dimensional induction tool described in Rosthal et al., “Field Tests of an Experimental Fully Tri-axial Induction Tool”, SPWLA 44th Annual Logging Symposium, June, 2003, acquire large numbers of signals that must be corrected for the effects of the borehole prior to processing the corrected signals to obtain estimates of the formation conductivity tensor. 3-D induction tools are designed to have azimuthal, radial, and axial sensitivity so that the measured signals are sensitive to the conductivity anisotropy and the radial and axial conductivity distributions of the earth formations penetrated by the borehole. In a Cartesian co-ordinate system fixed in the logging sonde, the axial direction is along the direction of the sonde axis (i.e., borehole). Multiple depths of investigation are achieved by employing several (e.g. nine) receivers spatially separated in the axial direction from the transmitter. Each antenna coil has an associated magnetic dipole moment vector whose magnitude is proportional to the product of the cross-sectional area of the coil and the amplitude of the electrical current in the coil. The direction of the magnetic moment vector is normal to the plane of the coil. The directional sensitivity to the formation conductivity distribution is derived from antennas with magnetic dipole moments oriented in both the longitudinal (i.e., axial direction) and transverse directions. In an example of a 3D induction tool described hereinbelow, the transmitter comprises a tri-axial antenna (e.g., solenoidal coils with dipole moments in the longitudinal and two orthogonal transverse directions). The three shortest spacing receivers each have a single longitudinally oriented antenna whereas the six other receivers are tri-axial. The short spacing receivers acquire signals at a single frequency, whereas the six other receivers acquire signals at two frequencies. In induction logging the transmitter is energized by an alternating current that causes alternating currents to flow in the conductive formation and borehole surrounding the logging sonde. The currents induce voltages in the receiver coils that are in-phase (i.e., resistive) and ninety degrees out-of-phase (i.e., reactive) with respect to the transmitter current. The in-phase component is called the R-signal and the out-of-phase component is called the X-signal. A phase-sensitive detector is used to measure both components. A complex or phasor voltage can represent the R and X-channel signals. Each complex voltage includes two distinct measured signals. The set of measured receiver voltages are sensitive to the radial and axial distributions of formation conductivity, and the borehole signal. The measured signals need to be corrected to remove the effects of the borehole and therefore obtain a set of so-called borehole corrected signals. The borehole corrected signals would then be processed to determine, for example, the formation conductivity distribution and anisotropy.
The degree of difficulty of the borehole correction problem for the described type of 3-D induction tool surpasses that of prior art conventional induction tools. One reason is that the 3-D induction tool with several receivers acquires a much greater number of signals than prior tools. For example, the above 3-D induction tool acquires 234 signals (117 complex voltages) at each measured depth in the borehole, whereas a previous generation tool acquires approximately one-tenth as many signals. Transverse magnetic dipole (TMD) transmitter coils can create axial borehole currents that produce receiver signals with very large borehole effects, which is another reason why correcting the 3-D induction tool signals for borehole effects is more challenging than for conventional induction tools. TMD transmitter coils can excite long-range longitudinal (i.e., axial) currents in the borehole that can couple very strongly into the receiver coils. On the other hand, conventional induction tools have longitudinally oriented transmitter and receiver coils that produce borehole and formation currents that flow in planes transverse to the axial or borehole direction and therefore the receiver signals they excite have by comparison smaller borehole signals.
Aside from the large number of data channels, the limitations of prior art methods for computing borehole corrected signals for a 3-D induction tool stem also from the fact that the borehole signal for each data channel depends in a non-linear and complex fashion on numerous quantities including: borehole radius, mud conductivity, near wellbore formation conductivity, formation conductivity anisotropy factor, and tool position or standoff for an eccentered tool. For previous generation induction tools, the borehole signal did not typically depend on conductivity anisotropy or on the direction of the standoff.
The dependence of the borehole effect on the direction of the standoff for a TMD transmitter can be understood by a simple symmetry argument. A TMD coil excites circular current loops transverse to the direction of the dipole moment. By symmetry arguments, for a centered tool in a circular borehole, there is no net current flow in the direction of the receivers because there are equal and opposite currents flowing in the axial (borehole) direction. The same symmetry argument holds in circular boreholes if the coil is eccentered in a direction parallel to the dipole moment of the coil. In these cases the borehole effect on a received signal from a TMD transmitter is no worse than that from a coil with a longitudinal magnetic dipole (LMD) moment. The symmetry is broken, however, if the coil is displaced in a direction perpendicular to the dipole moment. In the latter case there is a net axial current that can strongly couple into the receiver coils and result in a larger borehole effect than that for a LMD transmitter having receiver coils at the same spacings. Consider the standoff direction for a TMD transmitter whose dipole moment is oriented along the x-direction. The direction of the standoff can be described by the unit vector, {circumflex over (n)}={circumflex over (x)} cos φ+ŷ sin φ, where φ is the azimuthal angle measured from the x-direction. A displacement of the dipole along the x-direction from the center of the borehole corresponds to φ=0 degrees whereas a displacement along the y-direction corresponds to φ=90 degrees. In general, the standoff direction and the magnitude of the standoff will correspond to an arbitrary and usually unknown value of the azimuthal angle.
If all of the aforementioned parameters upon which the borehole signal depends were known during logging operations, then a forward model consisting of a formation penetrated by a borehole could be used to invert the 3-D induction raw measurements and determine the formation electrical properties. This approach is not viable because some of the parameters upon which the borehole effect depends are typically either not known (e.g., conductivity anisotropy, standoff) or only known approximately. Alternatively, an inversion might be used to determine both borehole and formation properties. But the latter approach is not viable either, because the computations would be computationally too intensive to be performed in real time during logging operations.
Various techniques for implementing borehole correction are disclosed in the following prior documents: “Real-Time Environmental Corrections for the DIT-E Phasor Dual Induction Tool” by T. Barber published by the Soc. of Prof. Well Log Analysts, 26th Annual Logging Symposium, Paper EE, 1985; “Accurate Logging in Large Boreholes” by C. Kienitz et al. published by the Soc. of Prof. Well Log Analysts, 27th Annual Logging Symposium, Paper III, 1986; U.S. Pat. No. 5,041,975; and U.S. Pat. No. 6,381,542. The techniques of these documents have one or more of the following limitations: use of charting that is not viable for a 3D induction tool; parameterization based on less complex tool responses; difficult empirical estimation of parameters; and/or use of algorithms that exhibit impractical convergence.
It is among the objects of the present invention to overcome limitations of prior art techniques with regard to borehole correction, particularly in complex tools, and to provide a method that: (1) does not require knowledge of the borehole parameters, (2) does not require or use a forward model to invert the measured signals, and (3) can be used to predict borehole corrected signals in substantially real time.